Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $x \neq 0$. $p = \dfrac{x + 5}{x - 6} \times \dfrac{x^2 - 2x - 24}{2x + 10} $
Explanation: First factor the quadratic. $p = \dfrac{x + 5}{x - 6} \times \dfrac{(x - 6)(x + 4)}{2x + 10} $ Then factor out any other terms. $p = \dfrac{x + 5}{x - 6} \times \dfrac{(x - 6)(x + 4)}{2(x + 5)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ (x + 5) \times (x - 6)(x + 4) } { (x - 6) \times 2(x + 5) } $ $p = \dfrac{ (x + 5)(x - 6)(x + 4)}{ 2(x - 6)(x + 5)} $ Notice that $(x + 5)$ and $(x - 6)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ \cancel{(x + 5)}(x - 6)(x + 4)}{ 2\cancel{(x - 6)}(x + 5)} $ We are dividing by $x - 6$ , so $x - 6 \neq 0$ Therefore, $x \neq 6$ $p = \dfrac{ \cancel{(x + 5)}\cancel{(x - 6)}(x + 4)}{ 2\cancel{(x - 6)}\cancel{(x + 5)}} $ We are dividing by $x + 5$ , so $x + 5 \neq 0$ Therefore, $x \neq -5$ $p = \dfrac{x + 4}{2} ; \space x \neq 6 ; \space x \neq -5 $